Chapter 9

Relativity

Basic Problems Newtonian mechanics fails to describe

properly the motion of objects whose speeds approach that of light

Newtonian mechanics is a limited theory It places no upper limit on speed It is contrary to modern experimental results Newtonian mechanics becomes a specialized

case of Einstein’s special theory of relativity When speeds are much less than the speed of light

Newtonian Relativity To describe a physical event, a frame of

reference must be established The results of an experiment performed

in a vehicle moving with uniform velocity will be identical for the driver of the vehicle and a hitchhiker on the side of the road

Newtonian Relativity, cont. Reminders about inertial frames

Objects subjected to no forces will experience no acceleration

Any system moving at constant velocity with respect to an inertial frame must also be in an inertial frame

According to the principle of Newtonian relativity, the laws of mechanics are the same in all inertial frames of reference

Newtonian Relativity – Example

The observer in the truck throws a ball straight up It appears to move in

a vertical path The law of gravity

and equations of motion under uniform acceleration are obeyed

Newtonian Relativity – Example, cont.

There is a stationary observer on the ground Views the path of the ball thrown to be a parabola The ball has a velocity to the right equal to the

velocity of the truck

Newtonian Relativity – Example, conclusion The two observers disagree on the shape of

the ball’s path Both agree that the motion obeys the law of

gravity and Newton’s laws of motion Both agree on how long the ball was in the air All differences between the two views stem

from the relative motion of one frame with respect to the other

Views of an Event An event is some

physical phenomenon Assume the event

occurs and is observed by an observer at rest in an inertial reference frame

The event’s location and time can be specified by the coordinates (x, y, z, t)

Views of an Event, cont. Consider two inertial frames, S and S’ S’ moves with constant velocity, ,

along the common x and x’ axes The velocity is measured relative to S Assume the origins of S and S’ coincide

at t = 0

Galilean Transformation of Coordinates

An observer in S describes the event with space-time coordinates (x, y, z, t)

An observer in S’ describes the same event with space-time coordinates (x’, y’, z’, t’)

The relationship among the coordinates are x’ = x – vt y’ = y z’ = z t’ = t

Notes About Galilean Transformation Equations The time is the same in both inertial

frames Within the framework of classical

mechanics, all clocks run at the same rate The time at which an event occurs for an

observer in S is the same as the time for the same event in S’

This turns out to be incorrect when v is comparable to the speed of light

Galilean Transformation of Velocity Suppose that a particle moves through a

displacement dx along the x axis in a time dt The corresponding displacement dx’ is

u is used for the particle velocity and v is used for the relative velocity between the two frames

Speed of Light Newtonian relativity does not apply to

electricity, magnetism, or optics These depend on the frame of reference used

Physicists in the late 1800s thought light moved through a medium called the ether The speed of light would be c only in a special,

absolute frame at rest with respect to the ether Maxwell showed the speed of light in free

space is c = 3.00 x 108 m/s

Michelson-Morley Experiment First performed in 1881 by Michelson Repeated under various conditions by

Michelson and Morley Designed to detect small changes in the

speed of light By determining the velocity of the Earth

relative to the ether

Michelson-Morley Equipment Used the Michelson

interferometer Arm 2 is aligned along the

direction of the Earth’s motion through space

The interference pattern was observed while the interferometer was rotated through 90°

The effect should have been to show small, but measurable, shifts in the fringe pattern

Active Figure AF_0903 the michelson-morley

experiment.swf

Michelson-Morley Results Measurements failed to show any change in

the fringe pattern No fringe shift of the magnitude required was ever

observed The negative results contradicted the ether hypothesis They also showed that it was impossible to measure

the absolute velocity of the Earth with respect to the ether frame

Light is now understood to be an electromagnetic wave, which requires no medium for its propagation

The idea of an ether was discarded

Albert Einstein 1879 – 1955 1905

Special theory of relativity 1916

General relativity 1919 – confirmation

1920’s Didn’t accept quantum

theory 1940’s or so

Search for unified theory - unsuccessful

Einstein’s Principle of Relativity Resolves the contradiction between Galilean

relativity and the fact that the speed of light is the same for all observers

Postulates The principle of relativity: All the laws of physics

are the same in all inertial reference frames The constancy of the speed of light: The speed

of light in a vacuum has the same value in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light

The Principle of Relativity This is a sweeping generalization of the

principle of Newtonian relativity, which refers only to the laws of mechanics

The results of any kind of experiment performed in a laboratory at rest must be the same as when performed in a laboratory moving at a constant velocity relative to the first one

No preferred inertial reference frame exists It is impossible to detect absolute motion

The Constancy of the Speed of Light This is required by the first postulate Confirmed experimentally in many ways Explains the null result of the

Michelson-Morley experiment Relative motion is unimportant when

measuring the speed of light We must alter our common-sense notions

of space and time

Consequences of Special Relativity A time measurement depends on the

reference frame in which the measurement is made There is no such thing as absolute time

Events at different locations that are observed to occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first

Simultaneity In special relativity, Einstein abandoned the

assumption of simultaneity Thought experiment to show this

A boxcar moves with uniform velocity Two lightning bolts strike the ends The lightning bolts leave marks (A’ and B’) on the

car and (A and B) on the ground Two observers are present: O’ in the boxcar and O

on the ground

Simultaneity – Thought Experiment Set-up

Observer O is midway between the points of lightning strikes on the ground, A and B

Observer O’ is midway between the points of lightning strikes on the boxcar, A’ and B’

Simultaneity – Thought Experiment Results

The light reaches observer O at the same time He concludes the light has traveled at the same

speed over equal distances Observer O concludes the lightning bolts occurred

simultaneously

Simultaneity – Thought Experiment Results, cont.

By the time the light has reached observer O, observer O’ has moved

The signal from B’ has already swept past O’, but the signal from A’ has not yet reached him

The two observers must find that light travels at the same speed

Observer O’ concludes the lightning struck the front of the boxcar before it struck the back (they were not simultaneous events)

Simultaneity – Thought Experiment, Summary Two events that are simultaneous in one

reference frame are in general not simultaneous in a second reference frame moving relative to the first

That is, simultaneity is not an absolute concept, but rather one that depends on the state of motion of the observer In the thought experiment, both observers are

correct, because there is no preferred inertial reference frame

Simultaneity, Transit Time In this thought experiment, the

disagreement depended upon the transit time of light to the observers and does not demonstrate the deeper meaning of relativity

In high-speed situations, the simultaneity is relative even when transit time is subtracted out We will ignore transit time in all further discussions

Time Dilation A mirror is fixed to the ceiling

of a vehicle The vehicle is moving to the

right with speed v An observer, O’, at rest in the

frame attached to the vehicle holds a flashlight a distance d below the mirror

The flashlight emits a pulse of light directed at the mirror (event 1) and the pulse arrives back after being reflected (event 2)

Time Dilation, Moving Observer Observer O’ carries a clock She uses it to measure the time

between the events (∆tp) She observes the events to occur at the

same place ∆tp = distance/speed = (2d)/c

Time Dilation, Stationary Observer

Observer O is a stationary observer on the Earth He observes the mirror and O’ to move with speed v By the time the light from the flashlight reaches the

mirror, the mirror has moved to the right The light must travel farther with respect to O than

with respect to O’

Active Figure AF_0905 time dilation.swf

Time Dilation, Observations Both observers must measure the

speed of the light to be c The light travels farther for O The time interval, ∆t, for O is longer

than the time interval for O’, ∆tp

Time Dilation, Time Comparisons

Time Dilation, Summary The time interval ∆t between two events

measured by an observer moving with respect to a clock is longer than the time interval ∆tp between the same two events measured by an observer at rest with respect to the clock ∆t > ∆tp

This is known as time dilation

Factor Time dilation is not observed in our

everyday lives For slow speeds, the factor of is so

small that no time dilation occurs As the speed approaches the speed of

light, increases rapidly

Factor Table

Identifying Proper Time

The time interval ∆tp is called the proper time interval The proper time interval is the time interval

between events as measured by an observer who sees the events occur at the same point in space

You must be able to correctly identify the observer who measures the proper time interval

Time Dilation – Generalization If a clock is moving with respect to you, the

time interval between ticks of the moving clock is observed to be longer that the time interval between ticks of an identical clock in your reference frame

All physical processes are measured to slow down when these processes occur in a frame moving with respect to the observer These processes can be chemical and biological

as well as physical

Time Dilation – Verification Time dilation is a very real phenomenon

that has been verified by various experiments

These experiments include: Airplane flights Muon decay Twin Paradox

Time Dilation Verification – Muon Decays

Muons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron

Muons have a half-life of ∆tp = 2.2 µs when measured in a reference frame at rest with respect to them (a)

Relative to an observer on the Earth, muons should have a lifetime of

∆tp (b) A CERN experiment measured

lifetimes in agreement with the predictions of relativity

Airplanes and Time Dilation In 1972 an experiment was reported that

provided direct evidence of time dilation Time intervals measured with four cesium

clocks in jet flight were compared to time intervals measured by Earth-based reference clocks

The results were in good agreement with the predictions of the special theory of relativity

The Twin Paradox – The Situation A thought experiment involving a set of twins,

Speedo and Goslo Speedo travels to Planet X, 20 light years

from the Earth His ship travels at 0.95c After reaching Planet X, he immediately returns to

the Earth at the same speed When Speedo returns, he has aged 13 years,

but Goslo has aged 42 years

The Twins’ Perspectives Goslo’s perspective is that he was at

rest while Speedo went on the journey Speedo thinks he was at rest and Goslo

and the Earth raced away from him and then headed back toward him

The paradox – which twin has developed signs of excess aging?

The Twin Paradox – The Resolution Relativity applies to reference frames moving

at uniform speeds The trip in this thought experiment is not

symmetrical since Speedo must experience a series of accelerations during the journey

Therefore, Goslo can apply the time dilation formula with a proper time of 42 years This gives a time for Speedo of 13 years and this

agrees with the earlier result There is no true paradox since Speedo is not

in an inertial frame

Length Contraction The measured distance between two points

depends on the frame of reference of the observer

The proper length, Lp, of an object is the length of the object measured by someone at rest relative to the object

The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length This effect is known as length contraction

Active Figure AF_0908 length contraction.swf

Length Contraction – Equation

Length contraction takes place only along the direction of motion

Length Contraction, Final The observer who measures the proper

length must be correctly identified The proper length between two points in

space is always the length measured by an observer at rest with respect to the points

Proper Length vs. Proper Time The proper length and proper time

interval are defined differently The proper length is measured by an

observer for whom the end points of the length remained fixed in space

The proper time interval is measured by someone for whom the two events take place at the same position in space

Lorentz Transformation Equations, Set-Up

Assume the event at point P is reported by two observers

One observer is at rest in frame S

The other observer is in frame S’ moving to the right with speed v

Lorentz Transformation Equations, Set-Up, cont. The observer in frame S reports the event

with space-time coordinates of (x, y, z, t) The observer in frame S’ reports the same

event with space-time coordinates of (x’, y’, z’, t’)

If two events occur, at points P and Q, then the Galilean transformation would predict that x = x’ The distance between the two points in space at

which the events occur does not depend on the motion of the observer

Lorentz Transformations Compared to Galilean The Galilean transformation is not valid

when v approaches c x = x’ is contradictory to length

contraction The equations that are valid at all

speeds are the Lorentz transformation equations Valid for speeds 0 v < c

Lorentz Transformations, Equations To transform coordinates from S to S’ use

These show that in relativity, space and time are not separate concepts but rather closely interwoven with each other

To transform coordinates from S’ to S use

Lorentz Velocity Transformation The “event” is the motion of the object S’ is the frame moving at v relative to S In the S’ frame

Lorentz Velocity Transformation, cont.

The term v does not appear in the u’y and u’z equations since the relative motion is in the x direction

When v is much smaller than c, the Lorentz velocity transformations reduce to the Galilean velocity transformation equations

If v = c, u’x = c and the speed of light is shown to be independent of the relative motion of the frame

Lorentz Velocity Transformation, final

To obtain ux in terms of u’x, use

Relativistic Linear Momentum To account for conservation of momentum in

all inertial frames, the definition must be modified to satisfy these conditions

The linear momentum of an isolated particle must be conserved in all collisions

The relativistic value calculated for the linear momentum of a particle must approach the classical value as the particle’s speed approaches zero

is the velocity of the particle, m is its massu

Relativistic Form of Newton’s Laws The relativistic force acting on a particle whose

linear momentum is is defined as This preserves classical mechanics in the limit of

low velocities It is consistent with conservation of linear

momentum for an isolated system both relativistically and classically

Looking at acceleration it is seen to be impossible to accelerate a particle from rest to a speed u c

Speed of Light, Notes The speed of light is the speed limit of

the universe It is the maximum speed possible for

matter, energy and information transfer Any object with mass must move at a

lower speed

Relativistic Kinetic Energy The definition of kinetic energy requires

modification in relativistic mechanicsThe work done by a force acting on the

particle is equal to the change in kinetic energy of the particle The initial kinetic energy is zero

The work will be equal to the relativistic kinetic energy of the particle

Relativistic Kinetic Energy, cont Evaluating the integral gives

At low speeds, u << c, this reduces to the classical result of K = 1/2 m u2

Total Relativistic Energy

E = mc2 =K+ mc2 = K + ER

The term mc2 = ER is called the rest energy of the object and is independent of its speed

The term mc2 is the total energy, E, of the object and depends on its speed and its rest energy

Replacing , this becomes

Relativistic Energy – Consequences A particle has energy by virtue of its

mass alone A stationary particle with zero kinetic

energy has an energy proportional to its inertial mass

This is shown by E = K + mc2

A small mass corresponds to an enormous amount of energy

Energy and Relativistic Momentum It is useful to have an expression

relating total energy, E, to the relativistic momentum, p E2 = p2c2

+ (mc2)2

When the particle is at rest, p = 0 and E = mc2 Massless particles (m = 0) have E = pc

Mass and Energy When dealing with particles, it is useful to

express their energy in electron volts, eV 1 eV = 1.60 x 10-19 J

This is also used to express masses in energy units mass of an electron = 9.11 x 10-31 kg = 0.511 Me Conversion: 1 u = 929.494 MeV/c2

More About Mass When using Conservation of Energy,

rest energy must be included as another form of energy storage

This becomes particularly important in atomic and nuclear reactions

General Relativity Mass has two seemingly different properties

A gravitational attraction for other masses, mg

Given by Newton’s Law of Universal Gravitation An inertial property that represents a resistance to

acceleration, mi

Given by Newton’s Second Law

Einstein’s view was that the dual behavior of mass was evidence for a very intimate and basic connection between the two behaviors

Elevator Example, 1 The observer is at rest

in a uniform gravitational field directed downward

He is standing in an elevator on the surface of a planet

He feels pressed into the floor, due to the gravitational force

Elevator Example, 2 Here the observer is

in a region where gravity is negligible

A force is producing an upward acceleration of a = g

The person feels pressed to the floor with the same force as in the gravitational field

Elevator Example, 3 In both cases, an

object released by the observer undergoes a downward acceleration of g relative to the floor

Elevator Example, Conclusions Einstein claimed that the two situations

were equivalent No local experiment can distinguish

between the two frames One frame is an inertial frame in a

gravitational field The other frame is accelerating in a

gravity-free space

Einstein’s Conclusions, cont. Einstein extended the idea further and

proposed that no experiment, mechanical or otherwise, could distinguish between the two cases

He proposed that a beam of light should be bent downward by a gravitational field The bending would be small A laser would fall less than 1 cm from the

horizontal after traveling 6000 km

Postulates of General Relativity All the laws of nature have the same

form for observers in any frame of reference, whether accelerated or not

In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference in the absence of gravitational effects This is the principle of equivalence

Implications of General Relativity Time is altered by gravity

A clock in the presence of gravity runs slower than one where gravity is negligible

The frequencies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequencies This has been detected in the spectral

lines emitted by atoms in massive stars

More Implications of General Relativity A gravitational field may be

“transformed away” at any point if we choose an appropriate accelerated frame of reference – a freely falling frame

Einstein specified a certain quantity, the curvature of time-space, that describes the gravitational effect at every point

Curvature of Space-Time The curvature of space-time completely

replaces Newton’s gravitational theory There is no such thing as a gravitational field

according to Einstein Instead, the presence of a mass causes a

curvature of time-space in the vicinity of the mass This curvature dictates the path that all freely

moving objects must follow

Testing General Relativity

General relativity predicts that a light ray passing near the Sun should be deflected in the curved space-time created by the Sun’s mass

The prediction was confirmed by astronomers during a total solar eclipse

Effect of Curvature of Space-Time Imagine two travelers moving on parallel

paths a few meters apart on the surface of the Earth, heading exactly northward

As they approach the North Pole, their paths will be converging

They will have moved toward each other as if there were an attractive force between them

It is the geometry of the curved surface that causes them to converge, rather than an attractive force between them

Black Holes If the concentration of mass becomes

very great, a black hole may form In a black hole, the curvature of space-

time is so great that, within a certain distance from its center, all light and matter become trapped

Trip to Mars Assume a spacecraft is traveling to

Mars at 104 m/s Ignoring the rules of significant figures,

=1.000 000 000 6 This indicates that relativistic

considerations are not important for this trip

Trip to Nearest Star To make it to the nearest star in a

reasonable amount of time, assume a travel speed of 0.99 c

The travel time as measured by an observer on earth is 4.2 years

The length is contracted to 0.59 ly Instead of 4.2 ly

The time interval is now 0.60 year

Problems With The Trip Technological challenge to build a spacecraft

capable of traveling at 0.99c The design of a safety system to ward about

running into asteroids, meteoroids or other pieces of matter

The aging problem similar to the twin paradox Assuming a round trip, 8.4 yr will have passed on

earth, but only 1.2 yr for the travelers This problem would be magnified by longer trips